Spin–spin correlation functions of the XXZ- Heisenberg chain in a magnetic field

Abstract

Using algebraic Bethe ansatz and the solution of the quantum inverse scattering problem, we compute compact representations of the spin-spin correlation functions of the XXZ-1 2 Heisenberg chain in a magnetic field. At lattice distance m, they are typically given as the sum of m terms. Each term n of this sum, n = 1, . . . m, is represented in the thermodynamic limit as a multiple integral of order 2n + 1; the integrand depends on the distance as the power m of some simple function. The root of these results is the derivation of a compact formula for the multiple action on a general quantum state of the chain of transfer matrix operators for arbitrary values of their spectral parameters.

Dynamical correlation functions of the spin- chain

“…In this section, we prove Theorem 3.1 using the method developed in [15] and [1]. In order to apply this method, we need first to reduce the computation of the generating function (3.6) to the evaluation of the expectation value of some product of twisted transfer matrices.…”

“…The remaining steps are quite standard (see [14], [15]). In the thermodynamic limit the distribution of the ground state parameters {λ} can be described by the spectral density ρ tot (λ).…”

“…Different ways to study the time-independent correlation functions of this model were proposed in the series of works (see e.g. [10,11,12,13,14,15,16,17,18,19,20,21]). As for the dynamical correlation functions, up to now, the only known exact results concern the case of free fermions ∆ = 0 [22,23,24,25,26,27,28,29].…”

“…Therefore the action of e iH (0) t reduces to the one of some product of transfer matrices. This enables us to apply directly the results of [15] and [1], where we computed the action of any product of transfer matrices on arbitrary states in a compact form. In the present case it leads to a master equation for the dynamical spin-spin correlation function g zz (m, t), or more precisely for its generating function (see Theorem 3.1).…”

“…In the thermodynamic limit M → ∞, the model exhibits three different regimes depending on the value of the anisotropy parameter ∆ and on the magnetic field h. In this limit, following [13,14,15,1], we obtain a multiple integral representation for the dynamical σ z correlation function in the massive (∆ > 1) and massless (−1 < ∆ ≤ 1) regimes. It is given as a sum of multiple integrals (see (6.16)) involving the time dependence through the factors e iEt , where E is the bare one-particle energy.…”